PV pro¬le Q(θ) or Q(y) is changed by a ¬nite increment δQ within some

¬nite mixing region y1 < y < y2 or θ1 < θ < θ2 , in such a way as to

¯

respect the integral relation (8.4). The dashed lines show the initial Q(y)

¯

pro¬le. In the case of the cartoon in Figure 8.2a, the pro¬le of δQ(y) is a

simple N-shape, having negative slope within the mixing region.

Imagine that the mixing somehow takes place without any wave mecha-

¯

nism being involved. The PV invertibility principle says that when the Q

pro¬le changes then the mean velocity pro¬le must change too, by δ¯φ say.

v

In the limiting case (8.7) the relevant inversion is trivial, b being constant;

for instance in the slab model it is simply

∞

¯y

δ¯φ (y) =

v δQ(˜) b d˜ .

y (8.10)

y

¯

For the N-shaped δQ(y) pro¬le, the shape of δ¯φ (y) is a simple parabola. For

v

¯

the δQ(y) pro¬le implied by Figure 8.2b, the shape of δ¯φ (y) is qualitatively

v

the same, the parabola-like shape given by the right-hand plot, Figure 8.2c.

These mean ¬‚ow changes show a net momentum de¬cit. Notice that

y2 y2 ¯

y1 δ¯φ (y)dy =

v y1 yδ Q(y)b dy (integrating by parts): the total momentum

¯

change, ignoring a constant factor ρ, is equal to the ¬rst moment of δQ(y).

¯

This is negative for the N-shaped δQ(y) pro¬le. The ¬rst moment, and the

momentum change itself, both have unambiguous meanings in virtue of the

y ¯

integral relation (8.4), which implies that y2 δQ(y)b dy = 0 in the present

1

limiting case (8.7), with b constant. So the momentum de¬cit is indeed a

¯

de¬cit whenever δQ(y) is such that the mixing event was indeed a mixing

¯

event, in the sense of weakening the gradient of Q within the mixing region

y1 < y < y2 .

The argument just presented can easily be generalized from the slab

Solar tachocline dynamics: eddy viscosity or anti-friction? 125

Fig. 8.2. Demonstration that rearrangement of PV substance by layerwise-two-dim-

ensional mixing on a strati¬cation surface S, within some latitude band y1 < y <

y2 , must entail momentum transport outside the band hence wavelike as well as

turbulent ¬‚uctuations. (This follows from PV invertibility, and does not require

accurate material invariance of Q.) The quantitative examples in plots (b) and (c)

are by courtesy of P. H. Haynes (personal communication); for full mathematical

details see Killworth & McIntyre (1985) and Haynes (1989). Plot (a) shows idealized

¯

Q distributions before and after mixing; (b) shows the same in an accurate slab-

model simulation, using DQ/Dt = 0 together with the inversion (8.7); (c) shows the

resulting mean momentum change, given by equation (8.10), whose pro¬le would

take a simple parabolic shape in the idealized case corresponding to (a).

geometry to the spherical geometry, replacing equation (8.10) by

θ

’1 ¯˜ ˜˜

δ¯φ (θ) = r (sin θ)

v δQ(θ) b sin θ dθ . (8.11)

0

We can also remove the restriction to the limiting case (8.7), reverting to

¬nite N 2 and Ri. In the most accurate versions it is necessary to rede¬ne the

¯

mean Q around latitude circles as a weighted ˜isentropic™ mean at constant

‘, i.e. following a strati¬cation surface S, with weighting function b, so

as to respect the integral relation (8.4). It is then convenient to switch

to ‘ as the vertical coordinate, as discussed under the heading ˜isentropic

coordinates™ in the atmospheric-science literature (e.g. Andrews et al. 1987).

The main conclusion, that layerwise-two-dimensional PV mixing produces

a momentum or angular momentum de¬cit, still holds good.†

It follows that “ whatever the purely turbulent (Austausch) stresses that

might be involved “ such turbulent stresses cannot satisfy the momentum

budget on their own. This point was made long ago by Stewart & Thomson

(1977) who, however, used it to claim that turbulent mixing scenarios like

† In fact the conclusion holds exactly on each surface S in any thought experiment in which the

initial and ¬nal states are axisymmetric. The vertical coupling represented by the ‚r term in

quasi-geostrophic theory is incapable, by itself, of transporting absolute angular momentum,

and so never enters the calculation.

McIntyre

126

those of Figures 8.2a,b cannot be realized. This overlooked the possibility

that ˜the problem of turbulence™ might have a wavelike aspect, allowing mo-

mentum to be exchanged between the mixing region and its surroundings.

To summarize, then, the implication in reality is that wave-induced mo-

mentum transport, not con¬ned to mixing regions such as y1 < y < y2 in

Figure 8.2, is an essential part of the picture “ essential to making sense of

the ¬‚uid dynamics as a whole. The turbulent mixing scenarios can in fact

be realized, but only in the presence of waves, which, in the stratospheric

case at least, are chie¬‚y Rossby waves.

The Stewartson“Warn“Warn model played an important role in develop-

ing the conceptual framework just sketched, by illustrating, with great pre-

cision, how everything works and ¬ts together in a particular set of idealized

thought experiments. We may note too that the same thought experiments

provide especially clear examples of anti-frictional behaviour.

In each case a shear ¬‚ow vφ ∝ y is disturbed by monochromatic Rossby

¯

waves generated by an undulating boundary located at positive y, outside

the domain of Figure 8.2. The graphs plotted in Figures 8.2b,c come from

one such thought experiment but are qualitatively similar to those from

all the others. In each case the surf zone or mixing zone surrounds the

location y = 0, a so-called ˜critical line™ where, by de¬nition, vφ coincides

¯

with the longitudinal phase speed of the Rossby waves. The phase speed

is retrograde relative to the mean ¬‚ow throughout y > 0, so that Rossby-

wave propagation is possible there. The ¬‚uid behaviour within the surf zone

is complicated and chaotic in most cases; for detailed examples and for a

de¬nitive and thorough analysis see Haynes (1989). The behaviour is anti-

frictional because the momentum transport that accounts for the momentum

de¬cit in the surf zone, equation (8.10) and Figure 8.2c, is everywhere against

the mean momentum gradient ‚¯φ /‚y > 0, through a positive correlation

v

between vy and vφ .

More generally, equations (8.10) and (8.11) show that in any thought ex-

periment starting with solid rotation on a given strati¬cation surface S, such

as the solid rotation observed below the Sun™s tachocline, the formation of

mixing regions like that in Figure 8.2 will drive the system away from solid

rotation. This general point is reinforced by the integral relation (8.4). Be-

cause b is positive de¬nite, (8.4) tells us at once that the only way to mix Q

to homogeneity on a strati¬cation surface S is to make Q zero everywhere “

a fantastically improbable state on a planet rotating like the Earth, or in a

star rotating like the Sun. Since real Rossby waves do break, and do mix Q,

they must be expected to do so imperfectly, mixing more strongly in some

places than in others and producing the characteristic spatial inhomogeneity

Solar tachocline dynamics: eddy viscosity or anti-friction? 127

that always seems to be observed, as illustrated by Figure 8.1. Again, the

e¬ect is to drive the system away from solid rotation.

To be sure, one can imagine a thought experiment in which the air on and

near the strati¬cation surface S begins by rotating solidly, and then has its

angular velocity uniformly reduced by breaking Rossby waves. The PV mix-

ing would have to be distributed in just such a way as to give a uniformly

reduced pole-to-pole latitudinal pro¬le of Q, keeping it precisely propor-

tional to cos θ. But the tailoring of a Rossby-wave ¬eld to do this would

be a more delicate a¬air than standing a pencil on its tip, and the natural

occurrence of such a wave ¬eld would be another fantastically improbable

thing. A su¬cient reason for its improbability is the positive feedback as-

sociated with PV mixing. As soon as some region begins to be mixed,